ABSTRACT

The Decreased Adjustment (DA) and Mean Squared Error Ratio (MSER) charts provide practitioners with a simple yet effective method of adjusting and controlling continuous flow processes. These charts feature less than 100% process adjustment, lack of model dependency, the removal of the effect of shifts and trends through adjustment and the detection of changes in the magnitude of random variation. All of the above are accomplished using simple Shewhart-like charts that provide feedback and suggest action similar to that of traditional Statistical Process Control (SPC). This paper, however, will focus primarily on the role of the decision maker in the adjustment/control situation. One of the most important outcomes of this research is the development of a methodology to provide these decision makers with the ability to trade-off the various aspects of performance of the DA-MSER charts based on management goals and objectives. This methodology is described in detail after a brief overview of the DA-MSER charts.

INTRODUCTION

The Decreased Adjustment-Mean Squared Error Ratio (DA-MSER) procedure is a practical methodology for dynamic process control of continuous processes when adjustments must be accomplished by a human operator. This paper focuses primarily on the options that the procedures provide decision makers, enabling them to trade-off aspects of performance of the control-monitoring strategy based on management goals and objectives. The DA-MSER methodology for controlling continuous processes has an interface to operators that is similar to familiar control charts but performs comparably, or superior, to existing methods and has been tested using process simulation techniques (D’Souza and Ferrell-1998). Since a control scheme in the real world is only effective when it performs well and is used correctly by operators, the DA-MSER methodology is both unique and valuable since it has the added bonus for decision makers to control adjustment/monitoring costs and effectiveness.

Process control is a vital component of any system because it has tremendous influence on both final product quality, the cost of processing, and maintenance costs. Whether the system produces items for retail sale like automobiles or chemicals, or performs a service like treating potable water, control is critical to success. In general, the basic functions of a controller are to detect changes in the process that requires adjustment and either make the adjustment or notify a human operator that such an adjustment is necessary and to leave the process alone in all other conditions. To perform these functions, a controller must monitor the process and use this information to determine when adjustment is necessary and act accordingly or determine no adjustment is needed and leave things the process alone.

Historically, automatic controllers (for e.g., PID controllers) were used to make the adjustments and it was assumed that the human operators would infer an assignable cause was present by other means, hopefully before the plant or equipment was damaged. More recently, researchers have suggested the integration of SPC and the strategy behind the automatic controllers (called Engineering Process Control, EPC) as a method of addressing the total problem. The methods suggested for EPC-SPC integration perform adequately when the assumptions under which they were developed are true; however, they have feasibility issues in practical, real-world implementation. Most of the methods in existence are complex, the requirements to implement them, as well as the demands on the operators, are quite complicated. They require adjustments to be made almost constantly for good performance, a questionable requirement in practice and their effectiveness dependent on how well the underlying model can be defined a priori or how frequently it can be updated in the methodology. Most importantly, they do not possess the ability to allow decision makers to intelligently assess the tradeoffs associated with different performance characteristics. Additional information about existing control-monitoring strategies and details of the performance of the DA-MSER procedures can be found in D’Souza and Ferrell (1998).

DA-MSER CHARTS

The chart developed for the reduced adjustment of continuous processes and aptly named the Decreased Adjustment (DA) chart is based on the Shewhart chart concept. If the adjustment procedure is accurately controlling the process mean to the target, the only variable component in a process observation should be random error and therefore, adjustment is not necessary. In Shewhart charts, the 1s, 2s and 3s control limits encompass 68%, 95% and 99.7% of the random error values. The same principle is used in the DA chart. If an observation falls outside the 3 s limits, there could be a change in the process mean and so an adjustment may be required. The practitioner could also choose to make adjustment decisions using 2s or 1s limits depending on the tradeoff between the costs associated with adjustment and those with not detecting a process change. The amount of adjustment to be made is a fraction of the deviation of an observation from the target as proposed by Grubbs (1983). The fraction decreases as adjustments are made on consecutive observations.

A sample DA chart is provided in Figure 1 below. The standard deviation of the random error component given as s_e is used to determine the width of the limits. These limits are called decision-making deadbands. In the chart given below, the multiplier for s _e is given as ‘k’. The value of ‘k’ is set on the basis of management objectives. The implications of different values of ‘k’ will be discussed in subsequent sections.

Figure 1. Decreased Adjustment (DA) Chart

The MSER chart was developed to facilitate the detection of changes in variation. It is based on the F-ratio concept. The MSER is a ratio of the mean squared deviation of the last ‘n’ observations from the historical or current process mean to the expected process variation determined through simulation. It can be defined as follows:

MSER = Mean Squared Deviation from historical/current mean for window of size ‘n’ Estimated process variation….. (1)

This MSER is plotted on the chart shown in Figure 2. The figure also contains 7 zones. The structure of the chart shown is also similar to that of a regular SPC control chart as well as the DA chart. The alpha error for the middle two regions (labeled Zone 3) is the same as for m ± 1s_e (a=0.32), for the middle three zones (labeled Zones 2, 3 and 4), the same as for m ± 2s_e (a=0.05) and finally, for Zones 1 through 5, as for m ± 3s_e (a = 0.003). But in this case, tabulated F values are used in determining the width of the zones. The lower bound of Zone1 is the F_{0.9985, n-1, ¥} value. Similarly, the lower bounds of Zones 2, 3, 4, 5 and 6 are the tabulated F_0.975,n-1,¥ , F_0.84,n-1,¥ , F_0.16,n-1,¥ , F_0.025,n-1,¥ and F_{0.0015,n-1,¥} values respectively.

If an MSER value falls in either Zone 0 or Zone 6, a change in process variation is indicated. Warnings are issued when MSER ratios fall in either Zone 1 or Zone 5 as shown in the figure below. As in the case of the deadband width ‘k’ of the DA chart, the window size ‘n’ of the MSER chart is determined by the decision maker in the control monitoring situation. The implications of different values of ‘n’ will also be discussed in the results section of this paper.

Figure 2. MSER Chart

PERFORMANCE MEASURES

For the evaluation of the DA-MSER procedures, four performance measures are defined. These measures are used to assess the various aspects of performance of the control-monitoring procedures. The controllable aspects of the DA-MSER procedures, which are deadband width ‘k’ for the DA chart and the window size ‘n’ of the MSER chart, can be determined by trading off the various aspects of performance. The measures are defined as follows:

Performance Measure 1: Mean or Bias

The mean of the adjusted values gives an indication of the effectiveness of the adjustment procedure. It provides an insight into the proximity of the adjusted process to a predetermined Target. The bias in the mean of the adjusted values is a derivative of the mean taking the Target value into consideration. It can be computed as:

Bias = |Process Mean - Target| ………. (2)

Performance Measure 2: Average Squared Deviation (ASD)

The ASD value provides insight into the amount of variability in a process as a result of an adjustment procedure. Therefore, the efficiency of the Decreased Adjustment procedure can be assessed using this measure, where:

ASD = 1/n S _i=1ⁿ (Observation - Target)² …..…… (3)

Performance Measure 3: Percentage Adjusted

Since reduced adjustment is an objective that is common to a large number of practitioners, this measure has been included to assess performance of the proposed procedures. The Percentage Adjusted measure can be defined as follows:

Percentage Adjusted = Number of adjustments made/Number of process observatns ……... (4)

Performance Measure 4: Average Run Length (ARL)

The ARL can be defined as the number of observations between the time that an assignable cause actually appears in a process input to when it is finally detected. It is desirable for monitoring procedures to have low ARL values.

CONSIDERATIONS FOR DECISION MAKERS IN CONTROL-MONITORING SITUATIONS

In the previous sections, control and monitoring procedures have been discussed along with measures to estimate their efficiency. The control procedures have four performance measures and it has been stated that they can be traded off against each other, depending upon the needs of the system and objectives of management.

In order to determine the values of the controllable factors, DA chart deadband width ‘k’ and MSER chart window size ‘n’, the performance of a control-monitoring strategy should be predictable. For this purpose, prediction models for three of the measures, Mean, ASD and Percentage Adjusted were developed in the course of this research. It is not feasible to develop a prediction model for ARL since it is based on shift magnitude, which is an uncontrollable factor. However, some heuristic methods for choosing controllable factors to minimize ARL are presented.

Messina, Montgomery, Keats and Runger (1996) present combinations of control and monitoring strategies based on cost considerations. They recommend combinations of control and monitoring procedures in the cases where it is cost-effective to have a low false alarm rate while the process is being monitored for changes in variation, when it is economical to have a high change detection rate and the case when no cost data is available.

The tradeoffs mentioned earlier involve the variability or ASD, the number of adjustments or percentage adjusted, the bias in the mean with respect to the Target, and the ARL. The choice of one scheme over another is dependent on the cost constraints and/or management policies. The choices can be made in the following way:

(1) If a high process change detection rate is required, the adjustment-monitoring scheme, which provides the smallest ARL for the amplitude and frequency combination of the process, is the best choice.

(2) If the cost considerations dictate as few adjustments as possible to reduce wear on equipment, the best combination with the lowest percentage adjusted is chosen.

(3) If the cost of being slightly off target is expensive, a combination where higher variability could be the sacrifice for smaller bias in the process mean is the procedure of choice.

(4) Similarly, if the cost of larger variation is much more than the cost of being slightly off target, a combination with the lowest ASD is chosen.

(5) For unavailable cost data, a combination that is acceptable to management with respect to all the 4 performance measures is chosen.

These five cost scenarios can be used to assess the ability of the developed procedure to provide an appropriate control-monitoring scheme for constrained processes.

EXPERIMENTAL INVESTIGATION

For the purpose of simulating different types of processes to determine the effect of the Decreased Adjustment procedure and determine rules for setting the deadbands, three sets of processes are considered. They are time series processes expressed in Fourier terms. The first has a single cosine component and the second has sine and cosine components each with the same frequency but different amplitudes. The last process also has sine and cosine components each with different amplitudes and frequencies. Additional information on these processes can be found in D’Souza and Ferrell (1998).

For the purpose of experimentation, the Target was set at 5 and the standard deviation of the random variation in the process (s_e) at 3. A preliminary or exploratory study was conducted involving 12 different amplitudes from 0.5 to 15 for each sine or cosine component and 8 frequencies ranging from 0.1 to 1. The deadband widths tested were 1s_e, 2s_e, 3s_e (that is, 3,6 and 9) units. The simulations were performed using all three models for 1000 replications, each replication lasting 500 time units.

Information from these three experiments was used to assess the behavior of the Decreased Adjustment procedure and to conduct another set of experiments to build prediction models. This time a 4⁴.3 (4 amplitudes, 4 frequencies, 2 sets of components, 3 deadband widths) experimental design was used. The simulations were run for 500 time units as before with 100 replications. The data from the experiment was used to construct a response surface for the prediction of Mean, ASD and Percentage Adjusted. The controllable factors for the experiments were the amplitude(s), frequency(s) and the deadband width.

The heuristics in the case of the MSER chart were developed by merely simulating the three processes as before for the 12 amplitude and 8 frequency combinations along with 5 window sizes (5, 10, 15, 20 and 30) which was a total of 1440 combinations. The behavior of the MSER ratio for the processes with no assignable causes was used to draw some conclusions as to how they could be used by management to make monitoring decisions for low ARL and false alarm levels.

RESULTS

Bias (or Mean)

The Means obtained through the experiment were not found to be predictable through the components of the process. This is primarily due to the fact that the means fall within a certain interval and the components of the process do not affect the mean to a large extent. Figure 3 given below indicates that the means are clustered around the Target (which is 5 and therefore Bias is 0). The only exception to this rule seems to be in the case where either of the amplitudes of the sine and cosine components are very large, the means stay pretty close to the Target value. It is safe to say that for the processes investigated with no change, a 95% confidence interval for the process mean can be stated as: 4.98232 ± 1.96 * 0.0133281 which is (4.9562, 5.0084).

Figure 3. Means in Prediction Model Experiment

Average Squared Deviation (ASD)

The ASD values were found to be highly predictable using a regression model, after the ASD was transformed using the natural logarithm transformation. Both amplitudes, frequencies, frequency squares, amplitude interactions and amplitude-frequency interactions were found to be highly significant in contributing to the final ASD value. The deadband width was not needed for prediction. This means that the ASD value for the DA procedure depends solely on process characteristics.

Percentage Adjustment

The percentage of adjustments made for a particular process if the dominant components and deadbands to be used were known, was found to be highly predictable as well. A square root transformation was used to perform a regression analysis and develop a prediction model. In the case of this performance measure, the regression equation can also be used in reverse to determine what deadbands should be used to maintain a particular percentage of time the process is adjusted.

All factors, interactions and square terms were found to be significant except for the frequency interaction. This means that not only do the process characteristics affect the percentage of adjustments but so does the deadband width. Within the constraints of the process dictated limits, deadbands can be set to attain a certain level of adjustment using the DA procedure.

Average Run Length

In order to use the suggested variance monitoring procedures, the number of MSER values falling in the extreme (or danger) zones, Zones 0 and 6, were counted. These values were compared to the ? error for the process.

Few combinations produced more observations in Zone 0 than the a (= 0.0015) value for the Zone. More than 60.2% of observations had a £ 0.0015 and more than 94.2% had a < 0.05. In the case of Zone 6, more than 70% of observations had a £ 0.0015 and more than 99.3% had a < 0.05. These values indicate that the false alarm rates for low for most process combinations monitored by the MSER procedure.

For Zone 0, the smallest sample size considered (i.e., 5) did the best with the number in the zone (Figure 4), which are in effect the False Alarms for the procedure. 65.28% of the observations with sample size 5 had a £ 0.0015 and all had a £ 0.0403.

Figure 4. Number of MSER values in Zone 0 by MSER Sample Size

The sample size of 10 does the best for Zone 6 (Figure 5) with nearly 70% of observations having a £ 0.0015 and all with a £ 0.0358.

Figure 5. Number of MSER values in Zone 6 by MSER Sample Size

Figure 6. ARL values for MSER Sample Size

In additional experiments to compare the performance of the MSER procedure with the monitoring procedures in existence as specified in D’Souza and Ferrell (1998), it was found that midsize sample sizes were more effective in detecting shifts in variation quickly as demonstrated in Figure 6. A tradeoff on sample size is necessary depending on the performance needs specified in the management objectives.

Results Summary

The prediction models for the performance measures and heuristics provide estimates that can be used by management to choose controllable factors like the MSER sample size and the maximum amount of adjustment feasible under cost constraints. Once these estimates are calculated and choices of controllable factors are made, for any set of management objectives, a viable control-monitoring procedure is available.

CONCLUSIONS

The results for this research were obtained through simulating processes, adjusting them through the Decreased Adjustment procedure, then monitoring for changes in variability through the MSER chart. The results presented prove that the deadband width for the DA adjustment chart and the window size of the MSER monitoring chart could be set by management based on the maximum amount of adjustment and the ARL and false alarm rate that could be tolerated as per their objectives.

All management objectives are not alike. Some of them are willing to trade a small amount of performance in one area to increase performance or cost-efficiency in another area. The DA-MSER control-monitoring scheme allows for such tradeoffs. The ability to accommodate these tradeoffs provides a valuable addition to EPC-SPC integration strategies.

REFERENCES

(1) D’Souza, A. J., Ferrell, W. G. (1998). A New Technique for the Dynamic Control of Continuous Flow Processes. In review for publication.

Grubbs, F. E. (1983). An Optimum Procedure for Setting Machines or Adjusting Processes. Journal of Quality Technology, October 1983, Vol. 15, No. 4, pp 186-189.

Messina, W. S., Montgomery, D. C., Keats, J. B., Runger, G. C. (1996). Strategies for Statistical Monitoring of Integral Control for the Continuous Process Industries. Statistical Applications in Process Control, edited by J. Bert Keats and Douglas C. Montgomery, Marcel Dekker, Inc.

(4) Montgomery, D. C. (1991). Statistical Quality Control. Second Edition. John Wiley and Sons.

Presented at the World Congress on Total Quality, January 7-8, 1999, Bombay, India